Bài này cũng dễ mà! Hãy cố gắng lên bạn sẽ làm được mà.

Con Ly Anh Tho kia biết trả lời thì trả lời đi chứ nói như vậy ai mà biết làm.

Ta có: \(A=\frac{2}{60\cdot63}+\frac{2}{63\cdot66}+\cdots+\frac{2}{117\cdot120}+\frac{2}{2003}\)

\(=\frac23\left(\frac{3}{60\cdot63}+\frac{3}{63\cdot66}+\cdots+\frac{3}{117\cdot120}\right)+\frac{2}{2003}\)

\(=\frac23\left(\frac{1}{60}-\frac{1}{63}+\frac{1}{63}-\frac{1}{66}+\cdots+\frac{1}{117}-\frac{1}{120}\right)+\frac{2}{2003}\)

\(=\frac23\left(\frac{1}{60}-\frac{1}{120}\right)+\frac{2}{2003}=\frac23\cdot\frac{1}{120}+\frac{2}{2003}=\frac{1}{60\cdot3}+\frac{2}{2003}=2\left(\frac{1}{360}+\frac{1}{2003}\right)\)

TA có: \(B=\frac{5}{40\cdot44}+\frac{5}{44\cdot48}+\cdots+\frac{5}{76\cdot80}+\frac{5}{2003}\)

\(=\frac54\left(\frac{4}{40\cdot44}+\frac{4}{44\cdot48}+\cdots+\frac{4}{76\cdot80}\right)+\frac{5}{2003}\)

\(=\frac54\left(\frac{1}{40}-\frac{1}{44}+\frac{1}{44}-\frac{1}{48}+\cdots+\frac{1}{76}-\frac{1}{80}\right)+\frac{5}{2003}\)

\(=\frac54\left(\frac{1}{40}-\frac{1}{80}\right)+\frac{5}{2003}=\frac54\cdot\frac{1}{80}+\frac{5}{2003}=5\left(\frac{1}{320}+\frac{1}{2003}\right)\)

\(A-B=\frac{2}{360}+\frac{2}{2003}-\frac{5}{320}-\frac{5}{2003}\)

\(=\frac{2}{360}-\frac{5}{320}-\frac{3}{2003}=\frac{2\cdot8-5\cdot9}{2880}-\frac{3}{2003}=\frac{-29}{2880}-\frac{3}{2003}<0\)

=>A<B

\(A=\frac{2}{60\cdot63}+\frac{2}{63\cdot66}+...+\frac{2}{117\cdot120}+\frac{2}{2003}\)

\(\text{Đặt }C=\frac{2}{60\cdot63}+\frac{2}{63\cdot66}+...+\frac{2}{117\cdot120}\)

\(C=\frac{2}{3}\left(\frac{3}{60\cdot63}+\frac{3}{63\cdot66}+...+\frac{3}{117\cdot120}\right)\)

\(C=\frac{2}{3}\left(\frac{1}{60}-\frac{1}{63}+\frac{1}{63}-\frac{1}{66}+...+\frac{1}{117}-\frac{1}{120}\right)\)

\(C=\frac{2}{3}\left(\frac{1}{60}-\frac{1}{120}\right)\)

\(C=\frac{2}{3}\cdot\frac{1}{120}\)

\(C=\frac{1}{180}\)

\(\text{Thay }C=\frac{1}{180}\text{Ta có : }\) \(A=\frac{1}{180}+\frac{2}{2003}\)

\(B=\frac{5}{40\cdot44}+\frac{5}{44\cdot48}+...+\frac{5}{76\cdot80}+\frac{5}{2003}\)

\(\text{Đặt }D=\frac{5}{40\cdot44}+\frac{5}{44\cdot48}+...+\frac{5}{76\cdot80}\)

\(D=\frac{5}{4}\left(\frac{4}{40\cdot44}+\frac{4}{44\cdot48}+...+\frac{4}{76\cdot80}\right)\)

\(D=\frac{5}{4}\left(\frac{1}{40}-\frac{1}{44}+\frac{1}{44}-\frac{1}{48}+...+\frac{1}{76}-\frac{1}{80}\right)\)

\(D=\frac{5}{4}\left(\frac{1}{40}-\frac{1}{80}\right)\)

\(D=\frac{5}{4}\cdot\frac{1}{80}\)

\(D=\frac{1}{64}\)

\(\text{Thay }D=\frac{1}{64}\text{ Ta có : }B=\frac{1}{64}+\frac{5}{2003}\)

\(\text{Vì }A=\frac{1}{180}+\frac{2}{2003}\text{ , }B=\frac{1}{64}+\frac{5}{2003}\)

\(\text{Có : }\frac{1}{180}< \frac{1}{64}\)

\(\frac{2}{2003}< \frac{5}{2003}\)

\(\Rightarrow\text{ }A< B\)

A=2/3.(3/60.63+3/63.66+.....+3/117.120+3/120.123)

A=2/3.(1/60-1/63+1/63-1/66+...+1/117-1/120+1/20-1/123)

A=2/3.(1/60-1/123)